The efficient recording of the trajectories of micron size particles that are moving throughout a three dimensional (3-D) space has been an important problem in several branches of science such as colloidal suspensions, the motion of algae or larvae in water, the motion of bacteria around and in cells, and the characterization of marine particulates.
One approach to tracking such particles has been the use of compound light microscopy. Conventional compound light microscopy can give high-resolution information about an object but only in a single focal plane. Therefore, compound light microscopy is effectively limited to two dimensional tracking of particles.
Digital in-line holography (DIH) offers a rapid and efficient approach to construct high-contrast 3-D images of a sample volume from a single hologram. An exemplary approach to DIH is described in U.S. Pat. No. 6,411,406 issued Jun. 25th, 2002 to H. Juergen Kreuzer which is incorporated in its entirety herein by reference. In digital in-line holography a spherical wave, emanating from a “point” source of linear dimensions of the order of the wavelength, illuminates an object, typically at a distance of a few thousand wavelengths from the source, and forms a highly magnified diffraction pattern on a screen much further away. Details of DIH and a thorough discussion of its history and potential have been presented in a number of publications together with earlier results in such diverse areas as cell biology, micro-particle imaging and tracking, and polymer crystallization.
Holography is a two-step process: first, a hologram must be recorded, and second, it must be reconstructed to yield an “image” of the object. In DIH the hologram is recorded by a detector array, such as a charge-coupled device (CCD) camera for detecting photons, and the frame (i.e. the detector recorded hologram) is then captured by a computer in which the reconstruction is done using numerical means. The role of reconstruction is to obtain the three-dimensional structure of the object from the two-dimensional hologram on the screen (i.e. the detector array), or, in physical terms, to reconstruct the wave front at the object. This can be achieved via a Kirchhoff-Helmholtz transform such as that represented in Equation 1.                               K          ⁡                      (                          r              →                        )                          =                              ∫            S                    ⁢                                                    ⅆ                3                            ⁢              ξ                        ⁢                                          I                ~                            ⁡                              (                                  ξ                  →                                )                                      ⁢                          exp              ⁡                              [                                  2                  ⁢                  π                  ⁢                                                                           ⁢                  i                  ⁢                                                            ξ                      →                                        ·                                                                  r                        →                                            /                      λξ                                                                      ]                                                                        (        1        )            
In the Kirchhoff-Helmholtz transform represented in Equation 1, the integration extends over the two dimensional surface of the screen (assumed to be perpendicular to the optical axis) with coordinates ξ=(X,Y,L), where L is the distance from the source (pinhole) to the center of the screen (detector array) and I(ξ) is the contrast image (hologram) on the screen, obtained by subtracting the images with and without the object present. The function K(r) is significantly structured and differs from zero only in the space region occupied by the object. By reconstructing the wave front K(r) on a number of planes at various distances from the source in the vicinity of the object, a three-dimensional image can be built up from a single two-dimensional hologram. K(r) is a complex function and one usually plots its magnitude to represent the object, although phase images are also available. For the numerical implementation of the transform a fast algorithm that evaluates K(r) without any approximations can be used. The algorithm employs a coordinate transformation that transforms the integral into a convolution that is solved by three consecutive Fast Fourier Transforms.
In holography, the term ‘reconstruction’ is used to describe obtaining the function K(r) from the hologram. The plot of |K(r)| on a two-dimensional plane perpendicular to the optical axis, is called a two dimensional (2-D) holographic reconstruction, is equivalent to a single in-focus image taken in a conventional compound microscope. In DIH a stack of 2-D holographic reconstructions can be generated from a single hologram. Combining such a stack results in a three-dimensional image of the object; this latter step is usually referred to as 3-D reconstruction.
The input to the 3-D reconstruction as represented, for example, by equation (1) is the contrast image for a perfectly spherical incoming wave. Perfecting this image is the hardest part in the practical implementation of DIH. The normal procedure for generating the contrast image is as follows: (i) Record digitally the hologram of the object, i.e. the intensity matrix, Inm, recorded on the detector array, where n and m enumerate pixels in the x- and y-axis. (ii) Remove the object and record digitally the intensity matrix of the illuminating laser. (iii) Subtract the results of (ii) from the results of (i) to numerically construct the contrast image, corrected for the intensity variations in the primary laser beam. Using this procedure almost all imperfections in the laser source are eliminated. Indeed, this procedure minimizes the quality requirements on the laser itself, as long as the laser is sufficiently stable to identically capture both images.
In many situations it may not be possible or convenient to remove the object from the optical path in order to construct the contrast image. This is clearly the case when the object is in motion and it is desired to record the time evolution of the object's trajectory.
What is needed is an effective approach for tracking the trajectories of particles and life forms that are moving throughout a three dimensional space.